Question

In Exercises 5-10, a list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) vertical shear by a factor of 2 (b) horizontal shear by a factor of 2

Short answer

The matrix is \( A = \begin{pmatrix} 5 & 2 \\ 2 & 1 \end{pmatrix} \).

Step-by-step solution

Understanding the Problem

We are asked to find a matrix that represents a sequence of transformations on the Cartesian plane. Specifically, we need to find a matrix that first performs a vertical shear by a factor of 2, followed by a horizontal shear by a factor of 2.

Matrix for Vertical Shear Transformation

A vertical shear transformation by a factor of 2 is represented by the matrix:\[A_1 = \begin{pmatrix} 1 & 0 \ 2 & 1 \end{pmatrix}\]This transformation keeps the x-coordinates the same and adds 2 times the x-coordinate to the y-coordinate.

Matrix for Horizontal Shear Transformation

A horizontal shear transformation by a factor of 2 is represented by the matrix:\[A_2 = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}\]This transformation keeps the y-coordinates unchanged and adds 2 times the y-coordinate to the x-coordinate.

Multiplying the Transformation Matrices

To apply these two transformations in sequence, we multiply their matrices in the order given:\[A = A_2 \times A_1 = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \ 2 & 1 \end{pmatrix}\]Performing the matrix multiplication, we calculate each element as follows:- First row, first column: \((1 \cdot 1) + (2 \cdot 2) = 1 + 4 = 5\)- First row, second column: \((1 \cdot 0) + (2 \cdot 1) = 0 + 2 = 2\)- Second row, first column: \((0 \cdot 1) + (1 \cdot 2) = 0 + 2 = 2\)- Second row, second column: \((0 \cdot 0) + (1 \cdot 1) = 0 + 1 = 1\)So the resulting transformation matrix is:\[A = \begin{pmatrix} 5 & 2 \ 2 & 1 \end{pmatrix}\]

Conclusion

The matrix that represents the sequence of transformations: first a vertical shear by a factor of 2 and then a horizontal shear by a factor of 2, is:\[A = \begin{pmatrix} 5 & 2 \ 2 & 1 \end{pmatrix}\]

Key concepts

Vertical Shear Transformation

In geometry, a vertical shear transformation is a linear transformation that shifts points vertically by a certain factor, while keeping the horizontal position (x-coordinate) unchanged. To understand this concept, imagine you have grid paper and you push the points vertically upwards or downwards but without altering their x-positions. This type of transformation is often used in computer graphics and animation.

When you perform a vertical shear by a factor of 2, each point on your Cartesian plane will have its y-coordinate altered as follows:

• The x-coordinate remains the same.
• The y-coordinate becomes: \( y' = y + 2x \)

This can be captured in matrix form using the transformation matrix:\[A_1 = \begin{pmatrix} 1 & 0 \ 2 & 1 \end{pmatrix}\]Here, any vector \((x, y)\) is transformed to \((x, y')\) using matrix multiplication, reinforcing the idea that only the y-coordinate is directly affected by the shear.

Horizontal Shear Transformation

A horizontal shear transformation is similar to a vertical shear but affects the x-coordinate instead. This transformation alters the x-coordinate of points across the Cartesian plane, shifting them horizontally and maintaining the y-coordinate constant. Visualize this as you apply a push on a horizontal axis, causing points to glide sideways.

• The y-coordinate stays unchanged.
• The x-coordinate transforms as \( x' = x + 2y \)

The matrix representation for a horizontal shear by a factor of 2 is:\[A_2 = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}\]In this matrix, the non-diagonal element '2' plays a crucial role in determining the amount of shear applied. When you multiply this matrix by a vector \((x, y)\), the result is a new vector \((x', y)\) where the x-coordinate reflects the shear, and the y-coordinate is left untouched.

Matrix Multiplication in Transformations

Matrix multiplication is a critical operation when dealing with sequences of transformations. This is because multiple transformations can be condensed into a single matrix through multiplication, allowing all transformations to be applied simultaneously. In this exercise, we have a vertical shear followed by a horizontal shear.

The transformation matrices are represented as \(A_1\) for the vertical shear and \(A_2\) for the horizontal shear. To find a single matrix \(A\) that performs both transformations in order, you multiply these two matrices:\[A = A_2 \times A_1 = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \ 2 & 1 \end{pmatrix}\]The resulting matrix \(A\) is calculated by multiplying corresponding rows and columns. Each element of the new matrix is derived by summing the products of corresponding elements:

• The element in the first row, first column: \( (1 \cdot 1) + (2 \cdot 2) = 5 \)
• In the first row, second column: \( (1 \cdot 0) + (2 \cdot 1) = 2 \)
• For the second row, first column: \( (0 \cdot 1) + (1 \cdot 2) = 2 \)
• Finally, the second row, second column: \( (0 \cdot 0) + (1 \cdot 1) = 1 \)

Thus, the final matrix that accomplishes both transformations is:\[A = \begin{pmatrix} 5 & 2 \ 2 & 1 \end{pmatrix}\]By understanding matrix multiplication, you realize that combining basic transformations is often more efficient and straightforward, making it easier to represent complex transformations as a single matrix.